- Email: [email protected]
Doppler-profile escape factors and escape probabilities for the cylinder and hemisphere
J. Quont. Spectrosc. Radior. $
Pergamon
PII: SOO22-4073(97)00033-2
Trons/ir Vol. 58. No. 3. pp. 347-354. 1997 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0022-4073/97 $17.00 + 0.00
DOPPLER-PROFILE ESCAPE FACTORS AND ESCAPE PROBABILITIES FOR THE CYLINDER AND HEMISPHERE A. K. BHATIAt Xaboratory for Astronomy MD 20771, U.S.A.
and S. 0. KASTNER$§
and Solar Physics, Goddard Space Flight Center. Greenbelt, and 11-A Ridge Road, Greenbelt, MD 20770. U.S.A. (Receired
12 November 1996)
Abstract-Escape
factors have been computed and tabulated for photons emitted within cylinders and hemispheres and absorbed/scattered according to the Doppler-profile function. The mean escape probability for Doppler-profile photons emerging normal to the axis of a cylinder has also been obtained. The assumptions under which these quantities are derived are described. Logistic-function approximations for the three types of escape probability are fitted to the tabulated values. Typographical errors were found and corrected in an earlier expression quoted in the literature. 0 1997 Elsevier Science Ltd
I. INTRODUCTION
The probability of direct (single-flight) photon escape from within a bounded radiating plasma, when averaged over the volume of the plasma and over all directions of emission, is conventionally referred to as the escape factor.‘.2 This quantity has been evaluated, in the monochromatic (single-frequency) case, for several source geometries including plane-parallel (slab) geometry, spherical geometry, and for some other configurations which were originally considered in connection with the topic of neutron diffusion [Case, deHoffman and Placzek3 (CDP)]. In the case of line emission over a range of frequencies, described by a frequency profile $(Y), in particular the Doppler (Gaussian) profile, the resulting mean escape factor averaged over the line profile appears to have been evaluated for only the plane-parallel and spherical geometries.4 However, some important astrophysical sources of radiation now being studied involve cylindrical geometry and even hemispherical geometry. The authors are presently engaged in spectroscopic modeling of the extreme ultraviolet (EUV) radiation from solar-active regions. Such regions contain individual radiating loops of cylindrical cross-section. When observed in higher-temperature EUV lines the radiating regions are further filled in, being approximately dome-like or hemispherical in form.’ In a first approximation, the statistical equilibrium equations describing atomic-level populations expected under optically thick conditions involve appropriate escape factors multiplying the radiative transition rates, so that the latter are effectively decreased. An early example of this approach is the system treated by Breton and Schwob.6 A more recent example is the system treated by the authors.7 We have therefore computed the Doppler-profile escape factors corresponding to cylindrical and hemispherical geometries, and present the results here. These include also monodirectional escape probabilities needed as factors in the emergent intensities for the cylindrical case. In the course of the work some critical typographical errors, found in a previous exposition (CDP) of the cylindrical case and reproduced in later work, have been noted and corrected. 2. ANALYSIS AND COMPUTATION The notation used here for single-flight escape probabilities is that of Kastner and Kastner;* i.e. the frequency-, direction- and space-averaged escape factor will be denoted by EF = p,(i,Rt,;l) where t0 is a characteristic optical thickness associated with the geometry. The Bartels parameter unity refers to the homogeneous or uniform excitation case. In the case of the cylinder of radius $To whom
all correspondence
should
be addressed. 347
A. K. Bhatiaand S. 0. Kastner
348
a and also the hemisphere of radius a, ?0 = k,,a where k,, is the line-center absorption coefficient. The latter quantity is given explicitly, in the case of the Doppler profile, by
k0 = (1.1612 x lo-“)&J;,
,
(1)
where & is the line-center wavelength in cm, AUis the line oscillator strength, M is the atomic or molecular weight in amu, T is the temperature, N, is the population of the lower level and FsE is a stimulated emission factor which may differ from unity. For brevity, we shall refer to the monochromatic and Doppler-profile cylinder escape factors respectively as CEF, and CEF,,, and to the corresponding hemisphere escape factors as HEF, and HEF,.
The computational procedure employed is to start with known monochromatic escape factors Doppler-profile function and HEF,,. and integrate these over the normalized it, where the dimensionless frequency variable x = (v v,,)/Av,, AvD being 4(x) = exp( - x2)/J the HW(l/e)M Doppler halfwidth. In the cylinder case, one Doppler-profile-averaged escape factor expression does exist in the literature6 and is cited first, though not directly made use of here. (a) Breton and Schwob6 obtained CEF, as the multiple integral: CEF,,
where the integrand is the monodirectional
frequency-integrated
(1 -exp[-
escape probability
(W
~~ exp( - x2)]} dx
and zL = 2koa cos(B)/sin($). Equation 2a can also be written as
where p, = COS(~), p0 E cos(8) and zL = 2koa4(l - P$)“~. Irons’ re-derived the same expression as his uniform excitation (“upper limit”) escape factor e,,, and gives a plot of 8,, in his Fig. 2, as did Breton and Schwab in their Fig. 1. The numerical accuracy required here proved rather difficult to achieve with this multiple integral, so that an equivalent and more convenient-though not as physically transparent-xpression for CEF, was made use of instead, as discussed next. (b) Two different expressions for the monochromatic cylinder escape factor, CEF,, are to be found in the literature. CDP described the chord method originated by Dirac which may be applied to obtain monochromatic escape factors for a variety of geometries. They gave a resulting expression [their Eq. (27)] for the cylinder escape factor CEF, in terms of modified Bessel functions Z,,(r) and K,,(z), which we have however found to contain typographical errors. The incorrect CDP expression has unfortunately also been reproduced in the text of Armstrong and Nicholls” and in the review’ by one of the present authors. The corrected expression is given here:
CEF, = ; z,
2[7&(7,)1,(7,,)
+
7,&(7,&(7,)
-
11 +
+ K,(7,.)&(7,)
-
Ko(7,,)4(7,,)
+
&(7t.)zo(L)
.
(3)
349
Doppler-profile escape factors and escape probabilities
The Doppler-profile
escape factor can then be obtained as the integral
s r
CEF,
=
--I
&,(x)CEF,.
(4)
dx ,
with r,. = 7,)exp( - x’). Equation (4) was evaluated here by using 20-point Gauss-Hermite integration, employing the Bessel function polynomial approximations given by Abramowitz and Stegun.” This proved to be more efficient computationally than the equivalent multiple-integral expression [Eq. 2~1. We mention, however, that some care must be taken in connection with the numerical use of these approximations. The separate functions Z”(z) and K,(r) vary with increasing t respectively as exp(r) and exp( - T), so that the products Z,(r)K,(r) are always to be computed. Also, the cross-over values of z differ between the polynomial approximations for the two functional types. CDP also gave (correct) approximations for CEF, in the limits r, -+O and r, -+ x. The monochromatic small-r expression may be integrated term-by-term over the Doppler profile to yield numerically for small TV: CEF,-+l
- (0.9428090)~~ + (0.1461035)ri
+ (0.2886751)~;
ln(2r,,)
(5)
This expression proved to be useful as a check on the computations. (c) For completeness we mention that a second equivalent expression for CEF, was given by Williams” without derivation, also in connection with neutron transport: CEF, = 2r,. ’ Z,(x)K,(x) dx/s’ sri
(6)
This expression was not evaluated here. 2. I. The hemisphere escape factor HEF,
The monochromatic escape factor for the hemisphere of radius a was derived by the chord method and given by CDP as: HEF,. = &
‘[I - exp( - 2rt_r)]ll/(_r)d.r , ’ s(I
where
$(y>
l/2)=
&[Jr;;(3+
f$)+
$
,os-lY]
(7b)
and
$(.YI
The hemisphere Doppler-profile
W)=$(Y>
I/2)-
% s
$($ - NY).
(7c)
escape factor can be obtained from this as: HEF,
=
-1.
ddx)HEF,
dx ,
(8)
with 5,.= x,, exp( - x2). This was computed with the MIDPNT subroutine’ for the inner integration of HEF,. and 20-point Gauss-Hermite integration for the outer integration.
350
A. K. Bhatia Table
Loe T” - 3.0 - 2.9 - 2.8 - 2.7 - 2.6 - 2.5 - 2.4 - 2.3 - 2.2 - 2.1 - 2.0 - 1.9 - 1.8 - 1.7 - I.6 - I.5 - 1.4 - 1.3 - I.2 - I.1 - I.0 - 0.9 - 0.8 - 0.7 - 0.6 - 0.5 - 0.4 - 0.3 - 0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1.0 I.1 1.2 1.3 1.4 I.5 I.6 I.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7
I. Escape
CEF,. monochromatic 0.99867 + 0.99833 + 0.99790 + 0.99735 + 0.99667 + 0.99582 + 0.99475 + 0.99340 -t 0.99172 + 0.98960 + 0.98697 + 0.98367 + 0.97956 + 0.97445 + 0.96810 + 0.96025 + 0.95056 + 0.93865 + 0.92409 + 0.90640 + 0.88502 + 0.85942 + 0.82905 + 0.79343 + 0.75220 + 0.70525 + 0.65275 + 0.59534 + 0.53412 + 0.47071 + 0.40715 + 0.34565 + 0.28836 + 0.23694 + 0.19239 + 0.15489 + 0.12403 + 0.98996 0.78865 0.62757 0.49906 0.39669 0.31524 0.25047 0.19899 0.15808 0.12558 0.99762 0.79256 0.62970 0.50035 0.39762 0.31603 0.25 123 0.19976 0.15889 0.12643 0.10065 0.80182 0.63934 0.51040 0.408 I2 0.32706 0.26291 0.21224 0.17235 0.14113 0.11690 -
and S. 0. Kastner
factors/probabilitiest
for the cylinder.
CEF,, Doooler 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3
0.99906 + 0.99882 + 0.99851 + 0.99813 + 0.99765 + 0.99704 + 0.99628 + 0.99532 + 0.99413 + 0.99263 + 0.99075 + 0.98840 + 0.98547 + 0.98181 + 0.97727 + 0.97163 + 0.96464 + 0.95603 + 0.94545 + 0.93252 + 0.91680 + 0.89782 + 0.87508 + 0.84811 + 0.81646 + 0.77981 + 0.73801 + 0.69117+0 0.63975 + 0.58461 + 0.52698 + 0.46844 + 0.41072 + 0.35552 + 0.30424 + 0.25785 + 0.21681 + 0.18113 + 0.15055 + 0.12462 + 0.10280 + 0.84572 0.69414 0.56848 0.46458 0.37891 0.30852 0.25092 0.20394 0.16569 0.13457 0.10921 0.88527 0.71656 0.57911 0.46743 0.37702 0.30408 0.24539 0.19820 0.16023 0.12963 0.10491 0.84912 0.68708 0.55588 0.44991 0.36471 -
CEPo monodirectionalt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3
0.99944 0.99930 0.99912 0.99889 0.99861 0.99825 0.99779 0.99722 0.99651 0.99560 0.99447 0.99305 0.99126 0.98902 0.98621 0.98269 0.97829 0.97280 0.96596 0.95746 0.94693 0.93396 0.91806 0.89867 0.87525 0.84723 0.81410 0.77551 0.73136 0.68187 0.62775 0.57017 0.51072 0.45128 0.39372 0.33963 0.29015 0.24587 0.20694 0.17318 0.14425 0.11968 0.98971 0.81610 0.67112 0.55045 0.45040 0.36781 0.29993 0.24436 0.19895 0.16186 0.13155 0.10678 0.86528 0.70004 0.56559 0.45655 0.36843 0.29735 0.24008 0.19391 0.15665 0.12653 0.10213 0.82339 0.66289 0.53288
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + -
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 3 3 3 *ontimed
Doppler-profile escape factors and escape probabilities
351
Table l-continued CEF,
Log RI 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0
0.98380 0.84583 0.14774 0.68428 0.65210 0.64946 0.67624 0.73386 0.82538 0.95568 0.11317 0.13628 0.16613 -
CEF, 4 4 4 4 4 4 4 4 4 4 3 3 3
0.29663 0.24266 0.20029 0.16745 0.14250 0.12416 0.11149 0.10385 0.10088 0.10246 0.10871 0.12002 0.13704 -
CEPD 3 3 3 3 3 3 3 3 3 3 3 3 3
0.42783 0.34320 0.27522 0.22073 0.17711 0.14221 0.11428 0.91903 0.73957 0.59535 0.47920 0.38548 0.30976 -
3 3 3 3 3 3 3 4 4 4 4 4 4
ta + b represents a x lo”. IEscape probability transverse to cylinder axis; see text.
CDP also reproduced an approximation for HEF,, in the limit ~“+0, which can be integrated term-by-term over the Doppler profile to give: HEFD(rO+O) z 1 -
(9a)
or numerically, HEF,
z 1 - (0.39774750)~~ + (0.1382930)r; - (0.04166666)~; + + (0.01103707)r~ .
This expression was similarly useful as a check on the computation
(9b)
of HEF,.
2.2. Escape probability normal to cylinder
Another escape probability needed in the modeling of radiation from an unresolved cylinder, such as a solar-active region loop, is the mean monodirectional escape probability fi,(D&,r,;l) for Doppler-profile photons emerging normal to the axis of the cylinder, which is the factor by which the (optically-thin-computed) normally emergent intensity must be decreased to account for opacity. Breton and Schwob6 dealt with a laboratory situation in which the cylindrical plasma was of course resolved, and therefore they could employ the particular monodirectional escape probability j,(D,i,r,,;l) associated with the diameter of the cylinder in their emergent intensity expression (7). Here we require the mean escape probability just mentioned, i.e., averaged across the cylinder, which is: $,(D,i,kOa;l)
= i
‘p, (D,i,2k,dm;l)
dy
,
(104
s0
p,GVb;U
= -&
s
‘, { 1
- exp[ - t exp( - x2)]} dx
(lob)
is the monodirectional escape probability associated with direction k and optical thickness r, so that the required escape probability (independent of cylinder radius a despite the apparent explicit dependence on this variable) is:
This escape probability was evaluated also, and is referred to below as CEP, (cylindrical escape probability).
352
A. K. Bhatia and S. 0. Kastner Table 2. Escape factors?for the hemisphere.
Log Gl -
3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 I.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.1 1.2 1.3 1.4 1.5 I.6 I.7 I.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7
HEF monochromatic
HEF, Doppler
0.99943+ 0 0.99929+ 0 0.99910+ 0 0.99887+ 0 0.99858+0 0.99822+ 0 0.99776+ 0 0.99718+ 0 0.99646+ 0 0.99554+ 0 0.99439+ 0 0.99295+O 0.99114+0 0.98887+O 0.98602+ 0 0.98244+0 0.97798+0 0.97239+O 0.96544+ 0 0.95678+ 0 0.94606+ 0 0.93282+0 0.91654+0 0.89667+ 0 0.87259+ 0 0.84366+0 0.80933+ 0 0.76913+0 0.72286+0 0.67068+ 0 0.61320+0 0.55163+ 0 0.48771+O 0.42358+0 0.36155+0 0.30370+ 0 0.25159+0 0.20610+0 0.16739+0 0.13511+o 0.10857+0 0.86977- I 0.69524- I 0.55485- I 0.44231- 1 0.35229- I 0.28041- I 0.22309- I 0.17742- I 0.14106- I 0.11213- I 0.89110-Z 0.70807- 2 0.56256- 2 0.44689- 2 0.35497- 2 0.28192- 2 0.22388-2 0.17776-2 0.14112-2 0.11202- 2 0.88897- 3 0.70531- 3 0.55944- 3 0.44359- 3 0.35159- 3 0.27855- 3 0.22057- 3
0.99960+ 0 0.99949+ 0 0.99937+o 0.99920+ 0 0.99900+ 0 0.99874+ 0 0.99841+0 0.99801+ 0 0.99749+ 0 0.99684+ 0 0.99603+ 0 0.99501+ 0 0.99373+o 0.99211+ 0 0.99009+ 0 0.98755+0 0.98438+0 0.98040+ 0 0.97544+ 0 0.96925+ 0 0.96156+0 0.95203+O 0.94027+ 0 0.92583+ 0 0.90819+ 0 0.88683+0 0.86119+ 0 0.83076+0 0.79514+ 0 0.75412+o 0.70777+ 0 0.65656+0 0.60138+ 0 0.54356+0 0.48473+ 0 0.42665+0 0.37100+0 0.31912+O 0.27192+0 0.22986+0 0.19301+o 0.16118+0 0.13398+0 0.11095+o 0.91584- I 0.75385- I 0.61891- I 0.50690- 1 0.41429- I 0.33801- I 0.27542- I 0.22422- I 0.18239- 1 0.14826- I 0.12038- I 0.97627-2 0.79054- 2 0.63922- 2 0.51624-2 0.41659-2 0.33607-2 0.27112-2 0.21878-2 0.17659-2 0.14253- 2 0.11500-2 0.92705- 3 0.74639- 3 continued
Doppler-profile
escape
factors
and escape
353
probabilities
Table 2-continued
Log To
HEFD
HEF,
3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0
0.17457 0.13807 0.10915 0.86235 0.68103 0.53770 0.42455 0.33534 0.26506 0.20973 0.16614 0.13175 0.10457
ta + b represents
-
3 3 3 4 4 4 4 4 4 4 4 4 4
0.60002 0.48159 0.38601 0.30909 0.24737 0.19796 0.15848 0.12695 0.10178 0.81672 0.65586 0.52693 0.42335
-
3 3 3 3 3 3 3 3 3 4 4 4 4
a x IOh.
3.
RESULTS
AND DISCUSSION
3.1. Results and approximating expressions Expressions (3), (4), (10) and (7), (8) for the respective escape factors/probabilities CEF,., CEF,, HEF,, were evaluated for - 3.0 5 log z,, I 5.0. The resulting values are given in Tables 1 and 2. For CEF, and HEF,,, similar but less complete tables were given in CDP. For use in computational applications such as the atomic models described above, it is desirable to represent the Doppler-profile probabilities CEFD, CEPD, HEF, by compact “analytical” expressions. A suitable representation for these quantities is afforded by the logistic function u(r,J = a/{ 1 + exp[b(log ?0 - c)]} with three parameters, a, b and c (the parameter a should not be confused with geometric diameter a). Fits to the three quantities were made, with the resulting parameter values being given in Table 3. These logistic parameters provide reasonable accuracy over the whole range tabulated; errors are within about + 1% up to T,,= 10 and about f 10% up to t,, = 100, increasing to about + 50% at very high optical thicknesses where probability values become very small in magnitude. CEPD and HEF,,
3.2. Assumptions
The two principal assumptions underlying the presently computed escape probabilities are the assumption of a spatially constant source function (“homogeneous source” or “uniform excitation”) within the cylinder or hemisphere, and the assumption of complete frequency redistribution (CFR). These assumptions must evidently be kept in mind when considering the applicability of the present results. High source densities/optical thicknesses will favor the second assumption but militate against the first assumption; at low source optical thicknesses the first assumption becomes more valid but, on the other hand, CFR gives way to partial frequency redistribution in which incident and scattered frequencies become correlated. Thus there is some optimum source optical thickness for which the escape probabilities computed here possess the most validity. 4. SUMMARY Escape factors have been computed and tabulated for photons emitted within cylinders and hemispheres of given optical thicknesses and absorbed/scattered according to the Doppler-profile function. The mean escape probability for Doppler-profile-distributed photons emerging normal
Table 3. Parameters Parameter
CEF,(o) 0.9999853 2.2952969 0.046747185
a
b C tAro)
for fit to logistic
5 ai{
1 + exp[b(logG - c)l}.
functiont. CEMr,)
I .0010796 2.3212136 0.22335545
HEF,(T”) 1.0010280 2.3024323 0.3787757
A. K. Bhatia and S. 0. Kastner
354
to the axis of a cylinder has also been obtained as a function of the optical thickness of the cylinder. The assumptions under which these quantities are derived are described. The escape factors and probabilities are applicable to problems involving cylindrical and hemispheric geometries, provided the constraints of the underlying assumptions are taken into account. Logistic-function approximations for the three types of escape probability are fitted to the tabulated values for convenient application. Also, long-standing typographical errors, which have propagated into later literature, were found and corrected in an expression given by Case, deHoffman and Placzek3 for the monochromatic cylinder escape factor. Acknowledgement-This
work was supported by NASA-RTOP grant 432-38-53-14.
REFERENCES 1. Irons, F. E., JQSRT, 2.
Kastner, S. O., Sp.
1979, 22, 1. Sci. Reo., 1994, 65, 317.
3. Case, K. M., deHoffman, F. and Placzek, G., Introduction to the Theory of Neutron Diffusion. U.S. Government Printing Office, Washington, D.C., 1953. 4. Capriotti, E. R., Astrophys. J., 1965, 142, 1101. 5. Bray, R. J., Cram, L. E., Durrant, C. J. and Loughhead, R.E., Plasma Loops in the Solar Corona. Cambridge University Press, Cambridge, 199 1. 6. Breton, C. and Schwab, J.-L., Compt. Rend. Acad. Sci. Paris, 1965, 260, 461. 7. Kastner, S. 0. and Bhatia, A. K., Astrophys. J. Suppl., 1989, 71, 665. 8. Kastner, S. 0. and Kastner, R. E., JQSRT, 1990, 44, 275. 9. Press, W. H., Flannery, B. P., Teukolsky S. A., and Vetterling, W. T., Numerical Recipes. Cambridge University Press, Cambridge, 1986. 10. Armstrong, B. H. and Nicholls, R. W., Emission, Absorption and Transfer of Radiation in Heated Atmospheres. Pergamon Press, Oxford, 1972. 11. Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions. U.S. Government Printing Office, Washington D.C., 1965. 12. Williams, M. M. R., Mathematical Methods in Transport Theory. Wiley Interscience, New York, 1971.
Pergamon
PII: SOO22-4073(97)00033-2
Trons/ir Vol. 58. No. 3. pp. 347-354. 1997 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0022-4073/97 $17.00 + 0.00
DOPPLER-PROFILE ESCAPE FACTORS AND ESCAPE PROBABILITIES FOR THE CYLINDER AND HEMISPHERE A. K. BHATIAt Xaboratory for Astronomy MD 20771, U.S.A.
and S. 0. KASTNER$§
and Solar Physics, Goddard Space Flight Center. Greenbelt, and 11-A Ridge Road, Greenbelt, MD 20770. U.S.A. (Receired
12 November 1996)
Abstract-Escape
factors have been computed and tabulated for photons emitted within cylinders and hemispheres and absorbed/scattered according to the Doppler-profile function. The mean escape probability for Doppler-profile photons emerging normal to the axis of a cylinder has also been obtained. The assumptions under which these quantities are derived are described. Logistic-function approximations for the three types of escape probability are fitted to the tabulated values. Typographical errors were found and corrected in an earlier expression quoted in the literature. 0 1997 Elsevier Science Ltd
I. INTRODUCTION
The probability of direct (single-flight) photon escape from within a bounded radiating plasma, when averaged over the volume of the plasma and over all directions of emission, is conventionally referred to as the escape factor.‘.2 This quantity has been evaluated, in the monochromatic (single-frequency) case, for several source geometries including plane-parallel (slab) geometry, spherical geometry, and for some other configurations which were originally considered in connection with the topic of neutron diffusion [Case, deHoffman and Placzek3 (CDP)]. In the case of line emission over a range of frequencies, described by a frequency profile $(Y), in particular the Doppler (Gaussian) profile, the resulting mean escape factor averaged over the line profile appears to have been evaluated for only the plane-parallel and spherical geometries.4 However, some important astrophysical sources of radiation now being studied involve cylindrical geometry and even hemispherical geometry. The authors are presently engaged in spectroscopic modeling of the extreme ultraviolet (EUV) radiation from solar-active regions. Such regions contain individual radiating loops of cylindrical cross-section. When observed in higher-temperature EUV lines the radiating regions are further filled in, being approximately dome-like or hemispherical in form.’ In a first approximation, the statistical equilibrium equations describing atomic-level populations expected under optically thick conditions involve appropriate escape factors multiplying the radiative transition rates, so that the latter are effectively decreased. An early example of this approach is the system treated by Breton and Schwob.6 A more recent example is the system treated by the authors.7 We have therefore computed the Doppler-profile escape factors corresponding to cylindrical and hemispherical geometries, and present the results here. These include also monodirectional escape probabilities needed as factors in the emergent intensities for the cylindrical case. In the course of the work some critical typographical errors, found in a previous exposition (CDP) of the cylindrical case and reproduced in later work, have been noted and corrected. 2. ANALYSIS AND COMPUTATION The notation used here for single-flight escape probabilities is that of Kastner and Kastner;* i.e. the frequency-, direction- and space-averaged escape factor will be denoted by EF = p,(i,Rt,;l) where t0 is a characteristic optical thickness associated with the geometry. The Bartels parameter unity refers to the homogeneous or uniform excitation case. In the case of the cylinder of radius $To whom
all correspondence
should
be addressed. 347
A. K. Bhatiaand S. 0. Kastner
348
a and also the hemisphere of radius a, ?0 = k,,a where k,, is the line-center absorption coefficient. The latter quantity is given explicitly, in the case of the Doppler profile, by
k0 = (1.1612 x lo-“)&J;,
,
(1)
where & is the line-center wavelength in cm, AUis the line oscillator strength, M is the atomic or molecular weight in amu, T is the temperature, N, is the population of the lower level and FsE is a stimulated emission factor which may differ from unity. For brevity, we shall refer to the monochromatic and Doppler-profile cylinder escape factors respectively as CEF, and CEF,,, and to the corresponding hemisphere escape factors as HEF, and HEF,.
The computational procedure employed is to start with known monochromatic escape factors Doppler-profile function and HEF,,. and integrate these over the normalized it, where the dimensionless frequency variable x = (v v,,)/Av,, AvD being 4(x) = exp( - x2)/J the HW(l/e)M Doppler halfwidth. In the cylinder case, one Doppler-profile-averaged escape factor expression does exist in the literature6 and is cited first, though not directly made use of here. (a) Breton and Schwob6 obtained CEF, as the multiple integral: CEF,,
where the integrand is the monodirectional
frequency-integrated
(1 -exp[-
escape probability
(W
~~ exp( - x2)]} dx
and zL = 2koa cos(B)/sin($). Equation 2a can also be written as
where p, = COS(~), p0 E cos(8) and zL = 2koa4(l - P$)“~. Irons’ re-derived the same expression as his uniform excitation (“upper limit”) escape factor e,,, and gives a plot of 8,, in his Fig. 2, as did Breton and Schwab in their Fig. 1. The numerical accuracy required here proved rather difficult to achieve with this multiple integral, so that an equivalent and more convenient-though not as physically transparent-xpression for CEF, was made use of instead, as discussed next. (b) Two different expressions for the monochromatic cylinder escape factor, CEF,, are to be found in the literature. CDP described the chord method originated by Dirac which may be applied to obtain monochromatic escape factors for a variety of geometries. They gave a resulting expression [their Eq. (27)] for the cylinder escape factor CEF, in terms of modified Bessel functions Z,,(r) and K,,(z), which we have however found to contain typographical errors. The incorrect CDP expression has unfortunately also been reproduced in the text of Armstrong and Nicholls” and in the review’ by one of the present authors. The corrected expression is given here:
CEF, = ; z,
2[7&(7,)1,(7,,)
+
7,&(7,&(7,)
-
11 +
+ K,(7,.)&(7,)
-
Ko(7,,)4(7,,)
+
&(7t.)zo(L)
.
(3)
349
Doppler-profile escape factors and escape probabilities
The Doppler-profile
escape factor can then be obtained as the integral
s r
CEF,
=
--I
&,(x)CEF,.
(4)
dx ,
with r,. = 7,)exp( - x’). Equation (4) was evaluated here by using 20-point Gauss-Hermite integration, employing the Bessel function polynomial approximations given by Abramowitz and Stegun.” This proved to be more efficient computationally than the equivalent multiple-integral expression [Eq. 2~1. We mention, however, that some care must be taken in connection with the numerical use of these approximations. The separate functions Z”(z) and K,(r) vary with increasing t respectively as exp(r) and exp( - T), so that the products Z,(r)K,(r) are always to be computed. Also, the cross-over values of z differ between the polynomial approximations for the two functional types. CDP also gave (correct) approximations for CEF, in the limits r, -+O and r, -+ x. The monochromatic small-r expression may be integrated term-by-term over the Doppler profile to yield numerically for small TV: CEF,-+l
- (0.9428090)~~ + (0.1461035)ri
+ (0.2886751)~;
ln(2r,,)
(5)
This expression proved to be useful as a check on the computations. (c) For completeness we mention that a second equivalent expression for CEF, was given by Williams” without derivation, also in connection with neutron transport: CEF, = 2r,. ’ Z,(x)K,(x) dx/s’ sri
(6)
This expression was not evaluated here. 2. I. The hemisphere escape factor HEF,
The monochromatic escape factor for the hemisphere of radius a was derived by the chord method and given by CDP as: HEF,. = &
‘[I - exp( - 2rt_r)]ll/(_r)d.r , ’ s(I
where
$(y>
l/2)=
&[Jr;;(3+
f$)+
$
,os-lY]
(7b)
and
$(.YI
The hemisphere Doppler-profile
W)=$(Y>
I/2)-
% s
$($ - NY).
(7c)
escape factor can be obtained from this as: HEF,
=
-1.
ddx)HEF,
dx ,
(8)
with 5,.= x,, exp( - x2). This was computed with the MIDPNT subroutine’ for the inner integration of HEF,. and 20-point Gauss-Hermite integration for the outer integration.
350
A. K. Bhatia Table
Loe T” - 3.0 - 2.9 - 2.8 - 2.7 - 2.6 - 2.5 - 2.4 - 2.3 - 2.2 - 2.1 - 2.0 - 1.9 - 1.8 - 1.7 - I.6 - I.5 - 1.4 - 1.3 - I.2 - I.1 - I.0 - 0.9 - 0.8 - 0.7 - 0.6 - 0.5 - 0.4 - 0.3 - 0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1.0 I.1 1.2 1.3 1.4 I.5 I.6 I.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7
I. Escape
CEF,. monochromatic 0.99867 + 0.99833 + 0.99790 + 0.99735 + 0.99667 + 0.99582 + 0.99475 + 0.99340 -t 0.99172 + 0.98960 + 0.98697 + 0.98367 + 0.97956 + 0.97445 + 0.96810 + 0.96025 + 0.95056 + 0.93865 + 0.92409 + 0.90640 + 0.88502 + 0.85942 + 0.82905 + 0.79343 + 0.75220 + 0.70525 + 0.65275 + 0.59534 + 0.53412 + 0.47071 + 0.40715 + 0.34565 + 0.28836 + 0.23694 + 0.19239 + 0.15489 + 0.12403 + 0.98996 0.78865 0.62757 0.49906 0.39669 0.31524 0.25047 0.19899 0.15808 0.12558 0.99762 0.79256 0.62970 0.50035 0.39762 0.31603 0.25 123 0.19976 0.15889 0.12643 0.10065 0.80182 0.63934 0.51040 0.408 I2 0.32706 0.26291 0.21224 0.17235 0.14113 0.11690 -
and S. 0. Kastner
factors/probabilitiest
for the cylinder.
CEF,, Doooler 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3
0.99906 + 0.99882 + 0.99851 + 0.99813 + 0.99765 + 0.99704 + 0.99628 + 0.99532 + 0.99413 + 0.99263 + 0.99075 + 0.98840 + 0.98547 + 0.98181 + 0.97727 + 0.97163 + 0.96464 + 0.95603 + 0.94545 + 0.93252 + 0.91680 + 0.89782 + 0.87508 + 0.84811 + 0.81646 + 0.77981 + 0.73801 + 0.69117+0 0.63975 + 0.58461 + 0.52698 + 0.46844 + 0.41072 + 0.35552 + 0.30424 + 0.25785 + 0.21681 + 0.18113 + 0.15055 + 0.12462 + 0.10280 + 0.84572 0.69414 0.56848 0.46458 0.37891 0.30852 0.25092 0.20394 0.16569 0.13457 0.10921 0.88527 0.71656 0.57911 0.46743 0.37702 0.30408 0.24539 0.19820 0.16023 0.12963 0.10491 0.84912 0.68708 0.55588 0.44991 0.36471 -
CEPo monodirectionalt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3
0.99944 0.99930 0.99912 0.99889 0.99861 0.99825 0.99779 0.99722 0.99651 0.99560 0.99447 0.99305 0.99126 0.98902 0.98621 0.98269 0.97829 0.97280 0.96596 0.95746 0.94693 0.93396 0.91806 0.89867 0.87525 0.84723 0.81410 0.77551 0.73136 0.68187 0.62775 0.57017 0.51072 0.45128 0.39372 0.33963 0.29015 0.24587 0.20694 0.17318 0.14425 0.11968 0.98971 0.81610 0.67112 0.55045 0.45040 0.36781 0.29993 0.24436 0.19895 0.16186 0.13155 0.10678 0.86528 0.70004 0.56559 0.45655 0.36843 0.29735 0.24008 0.19391 0.15665 0.12653 0.10213 0.82339 0.66289 0.53288
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + -
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 3 3 3 *ontimed
Doppler-profile escape factors and escape probabilities
351
Table l-continued CEF,
Log RI 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0
0.98380 0.84583 0.14774 0.68428 0.65210 0.64946 0.67624 0.73386 0.82538 0.95568 0.11317 0.13628 0.16613 -
CEF, 4 4 4 4 4 4 4 4 4 4 3 3 3
0.29663 0.24266 0.20029 0.16745 0.14250 0.12416 0.11149 0.10385 0.10088 0.10246 0.10871 0.12002 0.13704 -
CEPD 3 3 3 3 3 3 3 3 3 3 3 3 3
0.42783 0.34320 0.27522 0.22073 0.17711 0.14221 0.11428 0.91903 0.73957 0.59535 0.47920 0.38548 0.30976 -
3 3 3 3 3 3 3 4 4 4 4 4 4
ta + b represents a x lo”. IEscape probability transverse to cylinder axis; see text.
CDP also reproduced an approximation for HEF,, in the limit ~“+0, which can be integrated term-by-term over the Doppler profile to give: HEFD(rO+O) z 1 -
(9a)
or numerically, HEF,
z 1 - (0.39774750)~~ + (0.1382930)r; - (0.04166666)~; + + (0.01103707)r~ .
This expression was similarly useful as a check on the computation
(9b)
of HEF,.
2.2. Escape probability normal to cylinder
Another escape probability needed in the modeling of radiation from an unresolved cylinder, such as a solar-active region loop, is the mean monodirectional escape probability fi,(D&,r,;l) for Doppler-profile photons emerging normal to the axis of the cylinder, which is the factor by which the (optically-thin-computed) normally emergent intensity must be decreased to account for opacity. Breton and Schwob6 dealt with a laboratory situation in which the cylindrical plasma was of course resolved, and therefore they could employ the particular monodirectional escape probability j,(D,i,r,,;l) associated with the diameter of the cylinder in their emergent intensity expression (7). Here we require the mean escape probability just mentioned, i.e., averaged across the cylinder, which is: $,(D,i,kOa;l)
= i
‘p, (D,i,2k,dm;l)
dy
,
(104
s0
p,GVb;U
= -&
s
‘, { 1
- exp[ - t exp( - x2)]} dx
(lob)
is the monodirectional escape probability associated with direction k and optical thickness r, so that the required escape probability (independent of cylinder radius a despite the apparent explicit dependence on this variable) is:
This escape probability was evaluated also, and is referred to below as CEP, (cylindrical escape probability).
352
A. K. Bhatia and S. 0. Kastner Table 2. Escape factors?for the hemisphere.
Log Gl -
3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 I.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.1 1.2 1.3 1.4 1.5 I.6 I.7 I.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7
HEF monochromatic
HEF, Doppler
0.99943+ 0 0.99929+ 0 0.99910+ 0 0.99887+ 0 0.99858+0 0.99822+ 0 0.99776+ 0 0.99718+ 0 0.99646+ 0 0.99554+ 0 0.99439+ 0 0.99295+O 0.99114+0 0.98887+O 0.98602+ 0 0.98244+0 0.97798+0 0.97239+O 0.96544+ 0 0.95678+ 0 0.94606+ 0 0.93282+0 0.91654+0 0.89667+ 0 0.87259+ 0 0.84366+0 0.80933+ 0 0.76913+0 0.72286+0 0.67068+ 0 0.61320+0 0.55163+ 0 0.48771+O 0.42358+0 0.36155+0 0.30370+ 0 0.25159+0 0.20610+0 0.16739+0 0.13511+o 0.10857+0 0.86977- I 0.69524- I 0.55485- I 0.44231- 1 0.35229- I 0.28041- I 0.22309- I 0.17742- I 0.14106- I 0.11213- I 0.89110-Z 0.70807- 2 0.56256- 2 0.44689- 2 0.35497- 2 0.28192- 2 0.22388-2 0.17776-2 0.14112-2 0.11202- 2 0.88897- 3 0.70531- 3 0.55944- 3 0.44359- 3 0.35159- 3 0.27855- 3 0.22057- 3
0.99960+ 0 0.99949+ 0 0.99937+o 0.99920+ 0 0.99900+ 0 0.99874+ 0 0.99841+0 0.99801+ 0 0.99749+ 0 0.99684+ 0 0.99603+ 0 0.99501+ 0 0.99373+o 0.99211+ 0 0.99009+ 0 0.98755+0 0.98438+0 0.98040+ 0 0.97544+ 0 0.96925+ 0 0.96156+0 0.95203+O 0.94027+ 0 0.92583+ 0 0.90819+ 0 0.88683+0 0.86119+ 0 0.83076+0 0.79514+ 0 0.75412+o 0.70777+ 0 0.65656+0 0.60138+ 0 0.54356+0 0.48473+ 0 0.42665+0 0.37100+0 0.31912+O 0.27192+0 0.22986+0 0.19301+o 0.16118+0 0.13398+0 0.11095+o 0.91584- I 0.75385- I 0.61891- I 0.50690- 1 0.41429- I 0.33801- I 0.27542- I 0.22422- I 0.18239- 1 0.14826- I 0.12038- I 0.97627-2 0.79054- 2 0.63922- 2 0.51624-2 0.41659-2 0.33607-2 0.27112-2 0.21878-2 0.17659-2 0.14253- 2 0.11500-2 0.92705- 3 0.74639- 3 continued
Doppler-profile
escape
factors
and escape
353
probabilities
Table 2-continued
Log To
HEFD
HEF,
3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0
0.17457 0.13807 0.10915 0.86235 0.68103 0.53770 0.42455 0.33534 0.26506 0.20973 0.16614 0.13175 0.10457
ta + b represents
-
3 3 3 4 4 4 4 4 4 4 4 4 4
0.60002 0.48159 0.38601 0.30909 0.24737 0.19796 0.15848 0.12695 0.10178 0.81672 0.65586 0.52693 0.42335
-
3 3 3 3 3 3 3 3 3 4 4 4 4
a x IOh.
3.
RESULTS
AND DISCUSSION
3.1. Results and approximating expressions Expressions (3), (4), (10) and (7), (8) for the respective escape factors/probabilities CEF,., CEF,, HEF,, were evaluated for - 3.0 5 log z,, I 5.0. The resulting values are given in Tables 1 and 2. For CEF, and HEF,,, similar but less complete tables were given in CDP. For use in computational applications such as the atomic models described above, it is desirable to represent the Doppler-profile probabilities CEFD, CEPD, HEF, by compact “analytical” expressions. A suitable representation for these quantities is afforded by the logistic function u(r,J = a/{ 1 + exp[b(log ?0 - c)]} with three parameters, a, b and c (the parameter a should not be confused with geometric diameter a). Fits to the three quantities were made, with the resulting parameter values being given in Table 3. These logistic parameters provide reasonable accuracy over the whole range tabulated; errors are within about + 1% up to T,,= 10 and about f 10% up to t,, = 100, increasing to about + 50% at very high optical thicknesses where probability values become very small in magnitude. CEPD and HEF,,
3.2. Assumptions
The two principal assumptions underlying the presently computed escape probabilities are the assumption of a spatially constant source function (“homogeneous source” or “uniform excitation”) within the cylinder or hemisphere, and the assumption of complete frequency redistribution (CFR). These assumptions must evidently be kept in mind when considering the applicability of the present results. High source densities/optical thicknesses will favor the second assumption but militate against the first assumption; at low source optical thicknesses the first assumption becomes more valid but, on the other hand, CFR gives way to partial frequency redistribution in which incident and scattered frequencies become correlated. Thus there is some optimum source optical thickness for which the escape probabilities computed here possess the most validity. 4. SUMMARY Escape factors have been computed and tabulated for photons emitted within cylinders and hemispheres of given optical thicknesses and absorbed/scattered according to the Doppler-profile function. The mean escape probability for Doppler-profile-distributed photons emerging normal
Table 3. Parameters Parameter
CEF,(o) 0.9999853 2.2952969 0.046747185
a
b C tAro)
for fit to logistic
5 ai{
1 + exp[b(logG - c)l}.
functiont. CEMr,)
I .0010796 2.3212136 0.22335545
HEF,(T”) 1.0010280 2.3024323 0.3787757
A. K. Bhatia and S. 0. Kastner
354
to the axis of a cylinder has also been obtained as a function of the optical thickness of the cylinder. The assumptions under which these quantities are derived are described. The escape factors and probabilities are applicable to problems involving cylindrical and hemispheric geometries, provided the constraints of the underlying assumptions are taken into account. Logistic-function approximations for the three types of escape probability are fitted to the tabulated values for convenient application. Also, long-standing typographical errors, which have propagated into later literature, were found and corrected in an expression given by Case, deHoffman and Placzek3 for the monochromatic cylinder escape factor. Acknowledgement-This
work was supported by NASA-RTOP grant 432-38-53-14.
REFERENCES 1. Irons, F. E., JQSRT, 2.
Kastner, S. O., Sp.
1979, 22, 1. Sci. Reo., 1994, 65, 317.
3. Case, K. M., deHoffman, F. and Placzek, G., Introduction to the Theory of Neutron Diffusion. U.S. Government Printing Office, Washington, D.C., 1953. 4. Capriotti, E. R., Astrophys. J., 1965, 142, 1101. 5. Bray, R. J., Cram, L. E., Durrant, C. J. and Loughhead, R.E., Plasma Loops in the Solar Corona. Cambridge University Press, Cambridge, 199 1. 6. Breton, C. and Schwab, J.-L., Compt. Rend. Acad. Sci. Paris, 1965, 260, 461. 7. Kastner, S. 0. and Bhatia, A. K., Astrophys. J. Suppl., 1989, 71, 665. 8. Kastner, S. 0. and Kastner, R. E., JQSRT, 1990, 44, 275. 9. Press, W. H., Flannery, B. P., Teukolsky S. A., and Vetterling, W. T., Numerical Recipes. Cambridge University Press, Cambridge, 1986. 10. Armstrong, B. H. and Nicholls, R. W., Emission, Absorption and Transfer of Radiation in Heated Atmospheres. Pergamon Press, Oxford, 1972. 11. Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions. U.S. Government Printing Office, Washington D.C., 1965. 12. Williams, M. M. R., Mathematical Methods in Transport Theory. Wiley Interscience, New York, 1971.